On the unit group of some multiquadratic number fields
Let \(L=\prod_i k_i\) be the multiquadratic field generated as the composite of all its quadratic subfields \(k_i\). Denote by \(h(L)\) and \(E_L\) the class number and unit group of \(L\), respectively. \textit{S. Kuroda} [Nagoya Math. J. 1, 1--10 (1950; Zbl 0037.16101)] proved the formula \[ h(L) = {1\over{2^\nu}}q(L/{\mathbb Q})\prod_i h(k_i), \] where \(\nu\) has an explicit expression determined by the degree of \(L/{\mathbb Q}\) (and of different shape depending on whether \(L\) is real or complex) and where \(q = q(L/{\mathbb Q}) = (E_L\,:\,\prod_i E_{k_i})\), the so-called unit index. The authors study this index in the case that \(L\) is the genus field of a complex quadratic field \(k\) for which the 2-rank of the class group Cl\((k)\), say \(r_2\), is at most 3. A relatively easy result is that if \(r_2=1\) then \(q=1\). If \(r_2=2\), then a reasoning based on results of \textit{T. Kubota} [Nagoya Math. J. 10, 65--85 (1956; Zbl 0074.03001)] shows that \(q=8\) or 2 according as the 2-class field tower of \(k\) is of length 1 or \(>1\). The case \(r_2=3\) is more complicated, and the authors restrict to the situation that the 2-class group is of type \((2,2,2)\). It turns out that the result depends upon the rank of the Hilbert 2-class field \(k^1\) of \(k\), regarded as a module over the group ring \({\mathbb Z}[\text{{Gal}}(k^1/k)]\). This rank is either 2 or 3. In the former case \(q=2^7\). In the latter case the authors find conditions for fields with \(q=2^4\) and \(q=2^5\). They also present a numerical example of both cases.
scientific article; zbMATH DE number 5366207